# 6. Settlement of Shallow Footings

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6. Settlement of Shallow Footings
CIV4249: Foundation Engineering Monash University

Particular Sample Measurements: General Derived Relationship:
Oedometer Test Particular Sample Measurements: General Derived Relationship: (change of) Height Applied Load Void Ratio Applied Stress h

height vs time plots height ho elastic primary consolidation secondary
typically take measurements at 15s, 30s, 1m, 2m, 3m, 5m, 10m, 15m, 30m, 1h, 2h, 3h, 6h, 12h, 24h, 36h, 48h, 60h ….etc. elastic primary consolidation secondary compression typically repeat for 12.5, 25, 50, 100, 200, 400, 800 and 1600 KPa log time

Void ratio = f(h) e 1.00 e = 0.8 1 1 + e 1.917 2.65 Relative Volume
Specific Gravity h = 1.9 cm dia = 6.0 cm W = g 1 + e 1.917

Elastic Settlement By definition -
fully reversible, no energy loss, instantaneous Water flow is not fully reversible, results in energy loss, and time depends on permeability Clay Sand Instantaneous component Occurs prior to expulsion of water Undrained parameters Instantaneous component Expulsion of water cannot be separated Drained parameters Not truly elastic

Elastic parameters - clay
Eu Soft clay Firm clay Stiff Clay V stiff / hard clay Eu/cu most clays nu All clays kPa kPa kPa kPa 0.5 (no vol. change)

Elastic parameters - sand
Ed Loose sand Medium sand Dense sand V dense sand nd kPa kPa kPa kPa 0.1 to 0.3 0.3 to 0.4 note volume change!

Elastic Settlement E E ez r = H s/E = H.ez Q s H Generalized stress
and strain field r = ez .dz r = H s/E = H.ez

Distribution of Stress
Q Boussinesq solution e.g. sz = Q Is z2 y z R sz Is is stress influence factor r Is = 2p [1+(r/z)2]5/2 sr sq

load, q dr By integration of Boussinesq solution over complete area: dq a r z sz = q [ ] = q.Is [1+(a/z)2]3/2 sz

Stresses under rectangular area
Solution after Newmark for stresses under the corner of a uniformly loaded flexible rectangular area: Define m = B/z and n = L/z Solution by charts or numerically sz = q.Is B sz z Is = mn(m2+n2+1)1/2 . m2+n2+2 m2+n2-m2n2+1 4p m2+n2+1 + tan-1 2mn(m2+n2+1)1/2

Total stress change Is z/B

Computation of settlement
Q 1. Determine vertical strains: ß¥ â¥ 2. Integrate strains: y ez = 1 [sz - n ( sr + sq )] E ez = Q .(1+n).cos3y.(3cos2y-2n) 2pz2E z R sz r r = ez .dz sr r = Q (1-n2 ) prE sq

Settlement of a circular area
load, q Centre : dr dq r = 2q(1-n2).a E a r Edge : z r = 4q(1-n2).a pE sz

Settlement at the corner of a flexible rectangular area
Schleicher’s solution r = q.B 1 - n2 E Ir sz z m = L/B Ir = m ln ln 1 p 1+ m2 + 1 m m+ m2 + 1

Settlement at the centre of a flexible rectangular area
rcentre = 4q.B 2 1 - n2 E Ir Superposition for any other point under the footing

Settlement under a finite layer - Steinbrenner method
rcorner = q.B 1 - n2 E Ir Ir = F F2 1-n 1-2n q X Y B H E “Rigid”

Superposition using Steinbrenner method
L B

r = r(H1,E1) + r(H1+H2,E2) - r(H1,E2)
Multi-layer systems q r = r(H1,E1) + r(H1+H2,E2) - r(H1,E2) B H1 E1 E2 H2 “Rigid”

Primary Consolidation
A phenomenon which occurs in both sands and clays Can only be isolated as a separate phenomenon in clays Expulsion of water from soils accompanied by increase in effective stress and strength Amount can be reasonably estimated in lab, but rate is often poorly estimated in lab Only partially recoverable

Total stress change Is z/B

Pore pressure and effective stress changes
Ds = Du + Ds¢ At t = 0 : Ds = Du At t = ¥ : Ds = Ds¢ s¢f s¢i

Stress non-linearity qnet z

Soil non-linearity sv e s¢i s¢f p¢c r = S log + log p¢c s¢i s¢f Cr Cc
Cr H 1+eo Cc H 1+ec p¢c s¢i s¢f Cr Cc s¢i s¢f p¢c e sv

Coeff volume compressibility
r = Smv.Ds¢.DH (1+eo).mv e sv

Rate of Consolidation Flow h = H / 2 Flow h = H T = cv ti / H2
U = 90% : T = 0.848

Coefficient of Consolidation
Coefficient of consolidation, cv (m2/yr) Notoriously underestimated from laboratory tests Determine time required for (90% of) primary consolidation Why?

Secondary Compression
Creep phenomenon No pore pressure change Commences at completion of primary consolidation ca/Cc » 0.05 ca = De log (t2 / t1) r = log (t2/t1) caH (1+ep)

Flexible vs Rigid F F RF = 0.8 stress stress deflection deflection
rcentre RF = 0.8 0.8 rcentre

Depth Correction B z

rtot = RF x DF ( relas + rpr.con + rsec )
Total Settlement rtot = RF x DF ( relas + rpr.con + rsec )

Field Settlement for Clays (Bjerrum, 1962)

Differential Settlements
Guiding values Isolated foundations on clay < 65 mm Isolated foundations on sand <40 mm Structural damage to buildings 1/150 (Considerable cracking in brick and panel walls) For the above max settlement values flexible structure <1/300 rigid structure <1/500

Settlement in Sand via CPT Results (Schmertmann, 1970)